Eisenhart lift for higher derivative systems. (English) Zbl 1369.70041
Summary: The Eisenhart lift provides an elegant geometric description of a dynamical system of second order in terms of null geodesics of the Brinkmann-type metric. In this work, we attempt to generalize the Eisenhart method so as to encompass higher derivative models. The analysis relies upon Ostrogradsky’s Hamiltonian. A consistent geometric description seems feasible only for a particular class of potentials. The scheme is exemplified by the Pais-Uhlenbeck oscillator.
MSC:
| 70H50 | Higher-order theories for problems in Hamiltonian and Lagrangian mechanics |
| 70G60 | Dynamical systems methods for problems in mechanics |
| 37N05 | Dynamical systems in classical and celestial mechanics |
References:
| [1] | Ong, C. P., Curvature and mechanics, Adv. Math., 15, 269 (1975) · Zbl 0303.53032 |
| [2] | Eisenhart, L. P., Dynamical trajectories and geodesics, Ann. Math., 30, 591 (1929) · JFM 55.0452.06 |
| [3] | Duval, C.; Burdet, G.; Kunzle, H.; Perrin, M., Bargmann structures and Newton-Cartan theory, Phys. Rev. D, 31, 1841 (1985) |
| [4] | Brinkmann, H. W., Einstein spaces which are mapped conformally on each other, Math. Ann., 94, 119 (1925) · JFM 51.0568.03 |
| [5] | Gibbons, G. W.; Houri, T.; Kubiznak, D.; Warnick, C., Some spacetimes with higher rank Killing-Stackel tensors, Phys. Lett. B, 700, 68 (2011) |
| [6] | Gibbons, G. W.; Rugina, C., Goryachev-Chaplygin, Kovalevskaya, and Brdička-Eardley-Nappi-Witten pp-waves spacetimes with higher rank Stäckel-Killing tensors, J. Math. Phys., 52, 122901 (2011) · Zbl 1273.81068 |
| [7] | Galajinsky, A., Higher rank Killing tensors and Calogero model, Phys. Rev. D, 85, Article 085002 pp. (2012) |
| [8] | Cariglia, M.; Gibbons, G. W., Generalised Eisenhart lift of the Toda chain, J. Math. Phys., 55, Article 022701 pp. (2014) · Zbl 1298.37045 |
| [9] | Cariglia, M.; Gibbons, G. W.; van Holten, J. W.; Horváthy, P. A.; Kosinski, P.; Zhang, P. M., Killing tensors and canonical geometry, Class. Quantum Gravity, 31, 125001 (2014) · Zbl 1295.83010 |
| [10] | Cariglia, M.; Gibbons, G. W.; van Holten, J. W.; Horváthy, P. A.; Zhang, P. M., Conformal Killing tensors and covariant Hamiltonian dynamics, J. Math. Phys., 55, 122702 (2014) · Zbl 1308.70020 |
| [11] | Cariglia, M.; Galajinsky, A., Ricci-flat spacetimes admitting higher rank Killing tensors, Phys. Lett. B, 744, 320 (2015) · Zbl 1330.83038 |
| [12] | Filyukov, S.; Galajinsky, A., Self-dual metrics with maximally superintegrable geodesic flows, Phys. Rev. D, 91, Article 104020 pp. (2015) |
| [13] | Cariglia, M., Hidden symmetries of Eisenhart lift metrics and the Dirac equation with flux, Phys. Rev. D, 86, Article 084050 pp. (2012) |
| [14] | Cariglia, M.; Duval, C.; Gibbons, G. W.; Horváthy, P. A., Eisenhart lifts and symmetries of time-dependent systems, Ann. Phys., 373, 631 (2016) · Zbl 1380.81244 |
| [15] | Pais, A.; Uhlenbeck, G. E., On field theories with nonlocalized action, Phys. Rev., 79, 145 (1950) · Zbl 0040.13203 |
| [16] | Smilga, A., Benign versus malicious ghosts in higher-derivative theories, Nucl. Phys. B, 706, 598 (2005) · Zbl 1137.81355 |
| [17] | Duval, C.; Gibbons, G.; Horváthy, P., Celestial mechanics, conformal structures and gravitational waves, Phys. Rev. D, 43, 3907 (1991) |
| [18] | Duval, C.; Horváth, Z.; Horváthy, P. A., Vanishing of the conformal anomaly for strings in a gravitational wave, Phys. Lett. B, 313, 10 (1993) |
| [19] | Duval, C.; Horváthy, P. A.; Palla, L., Conformal properties of Chern-Simons vortices in external fields, Phys. Rev. D, 50, 6658 (1994) |
| [20] | Duval, C.; Horváth, Z.; Horváthy, P. A., The Nappi-Witten example and gravitational waves · Zbl 1021.81743 |
| [21] | Andrzejewski, K.; Gonera, J.; Machalski, P.; Maślanka, P., A note on the Hamiltonian formalism for higher-derivative theories, Phys. Rev. D, 82, Article 045008 pp. (2010) |
| [22] | Esen, O.; Guha, P., On geometry of Schmidt-Legendre transformation · Zbl 1411.70029 |
| [23] | Andrzejewski, K.; Galajinsky, A.; Gonera, J.; Masterov, I., Conformal Newton-Hooke symmetry of Pais-Uhlenbeck oscillator, Nucl. Phys. B, 885, 150 (2014) · Zbl 1323.70077 |
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